This follows from the Remark in [1]}, after the proof of Theorem 7. For the reader's convenience, we reproduce the short proof here. Let \(\delta >\frac{1}{2}\) . If \(d\)  is sufficiently large, we have \(\frac{d}{N-1}-\frac{1}{2}\ge \frac{1}{\sqrt{2}}\left(\delta -\frac{1}{2}\right)>0\) . Therefore,
\({\mathbb {P}}(\xi _{N-1}\ge d) &=& {\mathbb {P}}\left(\frac{\xi _{N-1}}{N-1}-\frac{1}{2}\ge \frac{d}{N-1}-\frac{1}{2}\right)\\&\le & {\rm exp}\,\left(-2\left(\frac{d}{N-1}-\frac{1}{2}\right)^2(N-1)\right)\) 
